Examples Kalman filter

The plant deseribed in Example 9.8 by equations (9.185) and (9.186) is to be eontrolled by a Linear Quadratie Gaussian (LQG) eontrol seheme that eonsists of a LQ Regulator eombined with the Kalman filter designed in Example 9.8. The... 

This tutorial uses the MATLAB Control System Toolbox for linear quadratie regulator, linear quadratie estimator (Kalman filter) and linear quadratie Gaussian eontrol system design. The tutorial also employs the Robust Control Toolbox for multivariable robust eontrol system design. Problems in Chapter 9 are used as design examples. 

In this chapter we discuss the principles of the Kalman filter with reference to a few examples from analytical chemistry. The discussion is divided into three parts. First, recursive regression is applied to estimate the parameters of a measurement equation without considering a systems equation. In the second part a systems equation is introduced making it necessary to extend the recursive regression to a Kalman filter, and finally the adaptive Kalman filter is discussed. In the concluding section, the features of the Kalman filter are demonstrated on a few applications. 

Equation (41.11) represents the (deterministic) system equation which describes how the concentrations vary in time. In order to estimate the concentrations of the two compounds as a function of time during the reaction, the absorbance of the mixture is measured as a function of wavelength and time. Let us suppose that the pure spectra (absorptivities) of the compounds A and B are known and that at a time t the spectrometer is set at a wavelength giving the absorptivities h (0- The system and measurement equations can now be solved by the Kalman filter given in Table 41.10. By way of illustration we work out a simplified example of a reaction with a true reaction rate constant equal to A , = 0.1 min and an initial concentration a , (0) = 1. The concentrations are spectrophotometrically measured every 5 minutes and at the start of the reaction after 1 minute. Each time a new measurement is performed, the last estimate of the concentration A is updated. By substituting that concentration in the system equation xff) = JC (0)exp(-A i/) we obtain an update of the reaction rate k. With this new value the concentration of A is extrapolated to the point in time that a new measurement is made. The results for three cycles of the Kalman filter are given in Table 41.11 and in Fig. 41.7. The... 

The above example illustrates the self adaptive capacity of the Kalman filter. The large interferences introduced at the wavelengths 26 and 28 10 cm have not really influenced the end result. At wavelengths 26 and 28 10 cm , the innovation is large due to the interfered. At 30 10 cm the innovation is high because the concentration estimates obtained in the foregoing step are poor. However, the observation at 30 10 cm is unaffected by which the concentration estimates are restored within the true value. In contrast, the OLS estimates obtained for the above example are inaccurate (j , = 0.148 and JCj = 0.217) demonstrating the sensitivity of OLS for model errors. 

The proposed technique is based on an extension to time-varying systems of Wiener s optimal filtering method (l-3). The estimation of the corrected chromato gram is optimal in the sense of minimizing the estimation error variance. A test for verifying the results is proposed, which is based on a comparison between the "innovations" sequence and its corresponding expected standard deviation. The technique is tested on both synthetic and experimental examples, and compared with an available recursive algorithm based on the Kalman filter ( ). 

Kalman filters can be extended to more complex situations with many variables and many responses. The model does not need to be multilinear but, for example, may be exponential (e.g. in kinetics). Although the equations increase considerably in complexity, the basic ideas are the same. 

Some Aspects of Parameters Identification in Mathematical Modelling 185 3.5.5.2 Example of the Use of the Kalman Filter... 

Figure 3.88 Numerical example of the Kalman filter tracking.

In this paper, the proposed approach to fault detection and isolation is a model-based approach. The first part of this communication focuses on the main fundamental concepts of the simulation library PrODHyS, which allows the simulation of the system reference model of a typical process example. Then, the proposed detection approach is presented. This exploits the extended Kalman Filter, in order to generate a fault indicator. In the last part, this approach is exploited through the simulation of the monitoring of a didactic example. 

One alternative to the direct online measurement of polymer properties is to use a process model in conjunction with optimal state estimation techniques to predict the polymer properties. Indeed, several online state estimation techniques such as Kalman filters, nonlinear extended Kalman filters (EKF), and observers have been developed and applied to polymerization process systems. ° In implementing the online state estimator, several issues arise. For example, the standard filtering algorithm needs to be modified to accommodate time-delayed offline measurements (e.g., MWD, composition, conversion). The estimation update frequency needs to be optimally selected to compensate for the model inaccuracy. Table 5 shows the extended Kalman filter algorithm with delayed offline measurements. Fig. 2 illustrates the use of online state estimator... 

These measurement points may be clustered for road surface detection and obstacle detection. Tracking these measurements over time, as implemented, for example by Kalman filtering, not only yields high reliability and accuracy, but also enables acquisition of information that is not directly observable in an image, such as speed or acceleration. 

Plate 3. Distribution of the nitrous oxide (N20) mixing ratio (ppbv) on the 10 hPa pressure level or approximately 30 km altitude as measured on November 6, 1994 by the CRISTA instrument on board the Space Shuttle. The data were interpolated by a Kalman filter. N20 can be regarded as a tracer of dynamical motions for example a tongue of Arctic air extrudes from the North polar vortex across the United States towards the Pacific Ocean. At the same time a "streamer" ofN20 rich air extends from the tropics across eastern Asia towards the Aleutian Islands and the west cost of Canada and the United States. Another "streamer" propagates along the eastern cost of the North America towards Europe. From Offermann et al J. Geophys. Res., 104, 16311, 1999. 

As demonstrated previously the process noise and the measurement noise parameters directly affect the state vectors estimated by the Kalman filter. Furthermore, the covariance matrix of the state estimation is affected as well. Therefore, accurate estimation of the noise parameters is necessary for good performance of the filter. In this example, the Bayesian approach is applied to select a p and a. Figure 2.32 shows the contours of the likelihood function p V 0, C) together with the actual noise variances 0 = [cr, and its optimal estimate 6. The two contours correspond to 50% and 10% of the peak value. The optimal values of ap = 2.8N and a = 7.1 x 10 m /s are at reasonable distance to the actual values as the actual noise variances are located within the region with significant probability density. Therefore, the Bayesian approach is validated to give accurate estimation for both noise variances for the linear oscillator. 

For implementation of the state estimation algorithm, it is necessary to define the estimator gain. Several state estimator algorithms have been proposed in the literature to calculate K. Table 8.4 highlights the most common state estimator algorithms with the respective strengths and weaknesses. Table 8.4 also presents some examples of implementations in polymerization reactors. For illustrative purposes the extended Kalman filter (EKF) will be briefly shown below. 

In this example, the matrix F is unstable (eigenvalues= 0,1 ), but = = Hy. Thus, the Kalman filter for this model converges to a steady state, and so the framework in this paper can still be used to devise and characterize appropriate classes of stationary inventory policies for the ARIMA(0,1,1) case. In fact, a detailed treatment of this model is provided in Graves (1999). 

An example of the adaptive SoC method using extended Kalman filter was studied by Plett [11-13]. Different cell models were investigated by the author and it was shown that the model with five state variables produces the best results [12]. By comparison, the model in Figure 15.11 has four states in total, including SoC and three additional ones (voltages of the capacitors). The results of SoC estimation showed that accuracy of 1% is possible for a presented specific test procedure. Moreover, the method was able to adapt... 

We will look at the details of each of these steps and the expected statistics of the results for a number of cases. We will start with the example of processing three LORAN TOAs (or two TDs), but the basic principles apply to GPS as well. We wiU then consider overdetermined solutions and the techniques for assuring the solutions are optimal in some sense. We will also consider integrated solutions and Kalman filter based solutions. 

Model calculations Perform the calculations or algorithms (step-by-step procedures for solving a problem) intended by a program, for example, process control, payroll calculations, or a Kalman Filter. 

Similar to other vision-based tracking systems [34], our system rehes on the visual tracking of a pattern composed of a set of features arranged in a known geometry. An example of a pattern would be a checkerboard with a known number of rows and columns, where the intersection of black and white squares serves as a feature. The particularity of our approach is that we do not require the detection of the whole pattern. Instead, we use a Kalman-filter based approach [38] to simultaneously perform... 

The Intel 8251, a Multibus example, a protocol adapter example, a block transfer example, a fifth-order digital elliptic wave filter example, a Kalman filter example, the BTL310, an ADPCM, the Risc-1, the MCS6502, and the IBM System/370. 

The Intel 8251, the MCS6502, the FRISC microprocessor, a Kalman filter example, a fifth-order digital elliptic filter... 

AFAP path-based scheduling, the Intel 8251, a Kalman filter example, a greatest common divisor algorithm example, a counter example, and others. Essentially an updated version of Camposano90a. 

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